Formula Sheets: Power System Stability | Power Systems - Electrical Engineering (EE) PDF Download

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P o w er Systems: P o w er System Stabilit y F orm ula
Sheet for Electrical GA TE
Swing Equation
• Swing Equation : Describ es rotor dynamics of a sync hronous mac hine.
2H
?
s
d
2
d
dt
2
=P
m
-P
e
-P
d
whereH is the inertia constan t (s ),?
s
is sync hronous sp eed (rads
-1
),d is rotor angle
(rad ), P
m
is mec hanical p o w er input, P
e
is electrical p o w er output, P
d
is damping
p o w er (all in p er unit or W ).
• Inertia Constan t :
H =
Stored kinetic energy at rated sp eed
Mac hine V A rating
=
1
2
J?
2
s
V A
(s)
where J is the momen t of inertia (kgm
2
).
• Small-Signal F orm : Linearized swing equation for small disturbances.
2H
?
s
d
2
?d
dt
2
=-?P
e
Steady-State Stabilit y
• P o w er-Angle Relationship : F or a t w o-mac hine system or mac hine connected to an
infinite bus.
P
e
=
EV
X
sind
whereE is the generator in ternal v oltage,V is the terminal or infinite bus v oltage, X
is the reactance, d is the p o w er angle.
• Sync hronizing P o w er Co e?icien t : Measure of steady-state stabilit y .
S =
dP
e
dd
=
EV
X
cosd
Stable if S > 0 .
• Maxim um P o w er T ransfer :
P
max
=
EV
X
T r ansien t Stabilit y
• Equal Area Criterion : F or stabilit y under a fault.
A
1
=A
2
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Page 2


P o w er Systems: P o w er System Stabilit y F orm ula
Sheet for Electrical GA TE
Swing Equation
• Swing Equation : Describ es rotor dynamics of a sync hronous mac hine.
2H
?
s
d
2
d
dt
2
=P
m
-P
e
-P
d
whereH is the inertia constan t (s ),?
s
is sync hronous sp eed (rads
-1
),d is rotor angle
(rad ), P
m
is mec hanical p o w er input, P
e
is electrical p o w er output, P
d
is damping
p o w er (all in p er unit or W ).
• Inertia Constan t :
H =
Stored kinetic energy at rated sp eed
Mac hine V A rating
=
1
2
J?
2
s
V A
(s)
where J is the momen t of inertia (kgm
2
).
• Small-Signal F orm : Linearized swing equation for small disturbances.
2H
?
s
d
2
?d
dt
2
=-?P
e
Steady-State Stabilit y
• P o w er-Angle Relationship : F or a t w o-mac hine system or mac hine connected to an
infinite bus.
P
e
=
EV
X
sind
whereE is the generator in ternal v oltage,V is the terminal or infinite bus v oltage, X
is the reactance, d is the p o w er angle.
• Sync hronizing P o w er Co e?icien t : Measure of steady-state stabilit y .
S =
dP
e
dd
=
EV
X
cosd
Stable if S > 0 .
• Maxim um P o w er T ransfer :
P
max
=
EV
X
T r ansien t Stabilit y
• Equal Area Criterion : F or stabilit y under a fault.
A
1
=A
2
1
where A
1
is the accelerating area (pre -fault to fault-on p erio d), A
2
is the decelerating
area (p ost-fault).
A
1
=
?
dc
d
0
(P
m
-P
fault
e
)dd, A
2
=
?
dm
dc
(P
p ost-fault
e
-P
m
)dd
where d
0
is i nitial angle, d
c
is clearing angle, d
m
is maxim um angle.
• Critical Clearing Angle :
cosd
c
=
1
P
p ost-fault
max
[
(P
m
(d
m
-d
0
)+P
fault
max
cosd
m
-P
p ost-fault
max
cosd
0
]
• Critical Clearing Time : Appro ximate using n umerical metho ds or swing equation
in tegration.
P o w er S ystem Stabilizers (PSS)
• PSS F unction : A dds damping to rotor oscillations via excitation con trol.
?P
e
=K
PSS
??
where K
PSS
is the stabilizer gain, ?? is the s p eed deviation.
Key Notes
• T ypical V alues : H ranges from 2 to 10 s for large mac hines.
• P er-Unit System : Use base MV A and base k V for calculations.
• Stabilit y Limits : System is stable if rotor angled < 90
?
for steady-state, and if equal
area criterion is satisfied for transien t stabilit y .
• Numerical Metho ds : Use R unge-Kutta or Euler metho ds for solving swing equation
in GA TE problems.
• A ssumptions: Neglect damping (P
d
= 0 ) unless s p ecified.
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